Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals
lemmalem:tensor-power-marginal-average-euclidean-2026aProbabilityFor a probability measure on R^q there is exactly one probability measure on R^{qN} giving each block rectangle the product of the factor masses; and for a probability measure on R^{qN} the average of its N block marginals is a probability measure on R^q integrating by averaging.Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts
lemmalem:particle-blocks-basic-euclidean-2026aAnalysisBlock maps are linear and Borel, inner products and norms on R^{qN} split over the particles, product maps act blockwise and inherit measurability, continuity and Lipschitz bounds, and diagonal points shift every particle.Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points
definitiondef:particle-blocks-euclidean-2026aAnalysisA point of R^{qN} is read as a configuration of N particles in R^q: the block maps extract the particles, configurations assemble them, product maps act particle by particle, and diagonal points repeat one particle.Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions
lemmalem:block-index-arithmetic-euclidean-2026bCombinatoricsThe numbers (k-1)q+i with k in [N] and i in [q], computed in the reals, are images of natural numbers that enumerate [qN] exactly once each, so a sum over [qN] splits into N blocks of q terms.Lebesgue Measure on ℝⁿ: σ-Finiteness, Bounded Sets, Open Sets and Coordinate Hyperplanes
lemmalem:lebesgue-measure-sigma-finite-euclidean-2026bAnalysisAssuming dependent choice, Lebesgue measure on ℝⁿ is σ-finite, gives bounded Borel sets finite measure and nonempty open sets positive measure, and every coordinate hyperplane is a null set.- A convex Lipschitz integrand with constant L is a convex function on the nonnegative reals vanishing at zero that is nondecreasing and Lipschitz with constant L.
Well-Posedness of the Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost: Existence and Uniqueness of a Bounded Viscosity Solution
theoremthm:langevin-density-cost-well-posed-wasserstein-2026bAnalysisProbabilityPDEFor a confining potential, positive noise, a common-noise matrix, control cost at most one, a bounded uniformly continuous running cost and the density cost of a convex Lipschitz integrand, in any dimension, the Langevin Hamilton-Jacobi equation with common noise and density cost…The Langevin Hamilton-Jacobi Operator with a Density Cost Satisfies the Hypotheses of the Comparison Principle and of Perron's Method
lemmalem:langevin-density-cost-comparison-hypotheses-wasserstein-2026cAnalysisProbabilityPDEFor a confining potential, positive noise, a common-noise matrix, control cost at most one, a bounded uniformly continuous running cost and the density cost of a convex Lipschitz integrand, the Langevin Hamilton-Jacobi operator with common noise and density cost is degenerate ell…The Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost on the Wasserstein Space
equationeq:langevin-density-cost-hamilton-jacobi-wasserstein-2026bAnalysisProbabilityPDEThe Langevin Hamilton-Jacobi equation in a confining potential with common noise along a common-noise matrix, with the density cost of a convex Lipschitz integrand added to the running cost.The Density Cost Along Optimal Maps is Controlled by the Monotonicity of the Score
lemmalem:density-cost-optimal-maps-euclidean-2026bAnalysisProbabilityFor two absolutely continuous measures with finite Fisher information and the optimal maps between them, the difference of their density costs is at most any positive multiple A of the sum of the pairings of the scores with the two optimal displacements, plus the dimension times…McCann's Jacobian Equation Along an Optimal Map Between Absolutely Continuous Measures
lemmalem:jacobian-equation-optimal-map-euclidean-2026aAnalysisProbabilityAlong the optimal map between two absolutely continuous measures with finite second moment, the density of the source equals the density of the target at the image point times the determinant of the Hessian of the convex potential, almost everywhere for the source.The Density Cost of a Convex Lipschitz Integrand
definitiondef:density-cost-euclidean-2026bAnalysisProbabilityThe density cost of a convex Lipschitz integrand assigns to an absolutely continuous probability measure with finite second moment the Lebesgue integral of the integrand composed with its density; it lies between zero and the Lipschitz constant.Well-Posedness of the Singular-Cost Dyson Hamilton-Jacobi Equation above the Hardy Threshold
theoremthm:dyson-singular-cost-well-posed-weyl-chamber-2026aAnalysisPDEFor a pair cost c times the sum of inverse squared gaps with c > -kappa^2/(8 theta), a harmonic confinement cost omega|x|^2 and bounded continuous g, the singular-cost Dyson equation on the Weyl chamber satisfies comparison in the class of u with u - P of P-subordinate growth, wh…- The Weyl chamber W_N is the set of points of R^N with strictly decreasing coordinates.
The Dyson Hamilton-Jacobi Equation in Singular-Cost Form on the Weyl Chamber
equationeq:dyson-singular-cost-hamilton-jacobi-weyl-chamber-2026aAnalysisPDEThe Hamilton-Jacobi equation lambda u + (theta/2)|Du|^2 - (kappa/2) tr D^2u = c S + omega |x|^2 + g on the Weyl chamber, with S the sum of the inverse squared gaps: a pair cost with reciprocal squares, a harmonic confinement cost and a running cost g, with no drift.Well-Posedness of the Dyson Hamilton-Jacobi Equation in the Weyl Chamber below the Collision Threshold
theoremthm:dyson-well-posed-weyl-chamber-2026aAnalysisPDEFor N particles with logarithmic repulsion of strength beta > kappa/2 and a convex confinement of at least linear and regular growth, and bounded continuous running cost g, the Dyson Hamilton-Jacobi equation on the Weyl chamber satisfies comparison in the class of P-subordinate g…The Dyson Hamilton-Jacobi Equation for N Controlled Particles in the Weyl Chamber
equationeq:dyson-hamilton-jacobi-weyl-chamber-2026aAnalysisPDEThe drift-form Hamilton-Jacobi equation on the Weyl chamber for N controlled particles with logarithmic repulsion of strength beta and confinement V_1: the penalty-drift equation with potential P = H_beta + sum V_1(x_k).The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality
lemmalem:confined-log-energy-weyl-chamber-2026aAnalysisFor P = H + sum V_1(x_k) on the Weyl chamber: P is C^2 with explicit gradient and Laplacian; it is a penalty when V_1 grows at least linearly; its gradient is monotone when V_1 is convex; and for kappa < 2 beta and convex V_1 of regular growth it satisfies the dissipation inequal…The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity
lemmalem:log-energy-weyl-chamber-basic-2026aAnalysisThe Weyl chamber is open; the logarithmic energy H is C^2 there with explicit gradient and Laplacian beta S, S the sum of inverse squared gaps; its gradient is monotone, admits a symmetrised form against any vector, satisfies the Calogero identity |DH|^2 = beta^2 S = beta tr D^2H…The Logarithmic Energy of N Ordered Particles on the Weyl Chamber
definitiondef:log-energy-weyl-chamber-2026aAnalysisThe logarithmic energy of strength beta on the Weyl chamber is H(x) = -beta times the sum over pairs i<j of log(x_i - x_j).